College

Would you classify 196 as a perfect square, perfect cube, both, or neither?

Choose the correct answer below.

A. The number is both a perfect square and a perfect cube because [tex]\square \times \square = 196[/tex] and [tex]\square \times \square \times \square = 196[/tex].

B. The number is a perfect cube because [tex]\square \times \square \times \square = 196[/tex].

C. The number is a perfect square because [tex]\square \times \square = 196[/tex].

D. The number is neither a perfect square nor a perfect cube because there is no integer that can be squared to get 196 and no integer that can be cubed to get 196.

Answer :

To classify the number, we follow these steps:

1. **Check for a Perfect Square:**

We look for an integer $n$ such that
$$ n^2 = 196. $$

Calculating the square root:
$$ \sqrt{196} = 14. $$

Since
$$ 14 \times 14 = 196, $$
the number $196$ is a perfect square.

2. **Check for a Perfect Cube:**

We look for an integer $m$ such that
$$ m^3 = 196. $$

Testing the values around the estimated cube root:
- $$ 5^3 = 125 $$
- $$ 6^3 = 216 $$

The number $196$ lies between $125$ and $216$, so there is no integer $m$ that satisfies $m^3 = 196$. Hence, $196$ is not a perfect cube.

3. **Conclusion:**

Since $196$ is a perfect square but not a perfect cube, the correct classification is that it is a perfect square.

Thus, the correct answer is:

$$\textbf{Option C: The number is a perfect square because } \square \times \square = 196.$$